Rechargeable batteries have become an essential part of modern electronics, for example, in smart phones, computers, cameras, cars, unmanned aerial vehicles, and other devices. For instance, lithium-ion (Li-ion) batteries are a popular choice as an energy storage medium due to their relatively large energy density. However, operating a Li-ion battery too aggressively can lead to a reduced cycle life and unpredictable thermal runaway reactions. Furthermore, charging of the Li-ion batteries may take too long time or be too inefficient for a specific purpose. These challenges reduce the usefulness of the Li-ion batteries.
In some instances, predictive physical models can be used to optimize the behavior of the rechargeable batteries. For example, based on the predictive physical models, battery charging/discharging can be selected such that, for example, the charging time is shorter, the temperature of the battery is lower, or the life of the battery is longer. With predictive physical models, often the representative system of equations includes a combination of ordinary differential equations (ODEs), partial differential equations (PDEs), and algebraic equations (AEs). The PDEs are often the governing equations of the system (e.g., a battery charging system) that vary in both space and time. When discretizing these PDEs spatially, the PDEs are reduced to a set of ODEs and AEs. This resulting combined set of ODEs and AEs is known as a set of differential algebraic equations (DAEs). In typical physical systems, the ODEs will often represent most of the governing equations, while the AEs act as constraints applied to the system to ensure that the solution accurately reflects the physical possibilities (e.g., conservation laws, boundary conditions, etc.).
For a system of DAEs, a set of consistent initial conditions (ICs) must be provided in order to solve the system with standard solvers. In some cases, even small deviations from consistent ICs will cause the DAE solver to fail. Therefore, some solvers have initialization routines that calculate consistent ICs from starting guesses. However, these routines add computational time and often require specific solvers to obtain the ICs used by the primary DAE solver.
Some conventional IC estimates use non-physical approximations such as setting differential variable gradients to zero initially. Since the true value of the ICs may be significantly different from the starting guess, this approach may create solver inefficiencies for the few cases that these solvers are able to solve. Furthermore, if the true value of the ICs is sufficiently different from the zero-gradient value, the DAE solver may fail.
Some conventional technologies calculate the ICs of differential and algebraic variables separately using an Euler backward step with a very small step size in order to obtain values very near the initial time. However, these conventional technologies also create solver inefficiencies due to the very small step size used to calculate the ICs, which requires a significant computational effort for calculating the ICs.
Other conventional technologies use successive linear programming, a Taylor series approximation, or a Laplace method to find ICs. However, these methods may also require a significant computational effort, or may produce ICs that are not sufficiently accurate for the DAE solver. Therefore, a need remains for battery charging/discharging technologies that are based on efficient predictive models.